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452

Beam Examples.

Beams of Uniform Strength.—If rectangular beams be
proportioned to their bending moment at every section, the depth
or width will vary as follows, easily proved by equation :—

With (
constant <
breadth f

With

constant

depth

Case I.           Depth    oc parabola.

„    II             „        a triangle.

nL> IV.    „        oc two parabolas.

;,   V.  '          ;;        cc

' Case I.           Breadth oc triangle.

„    II.               „       oc 2 convex parabolas.

„    III., IY.     „       cc 2 triangles.

3,    Y.               „       oc 2 concave parabolas.

Example 29.—A beam of oak, supported at the ends, 2' long, 2'
broad, 2" deep, supports 400 Ibs. safely, at the centre, and its de-
flection is -06". Find safe load at centre, and deflection of a beam
of oak 16' long, 9" broad, 14" deep, (i) with ends supported ; (2) with
ends fixed. (Eng-. Exam, 1882.)

& A2                        W I3

Taking / in feet and bh in ins.   W ex —   and A, ex -—

Sample "beam, W   oc

2X22

ex

. 400 x 5

"2x8"

200

TVT        1                     Tir            QX   I4X  14

Ne\v beam,     W1 oc   ~-----——- = 110-2

(i) Supported; W : Wl : : 4 : 110*2   and Wx =
11,020 x 4096

oc

9X2744

= 25,600

.*•   A :  A! :: 200 : 1828

,    .       i828x*o6        ft,,
and A^••

200

(2) Fixed; Bn(i) : Bm(2) : ;         ;

4        o

,'.   W2=r 1,020 x 2 = 22,040 Ibs.

and

Example 30.—A beam, of unifarrn section is supported at the ends
and loaded centrally. Find tne ratio of depth to spaa that the'deflec-
tion may not exceed j^W of span when,/== 8000 Ibs. and E =s 28,000,000.
(Rons. MacK Constr. Ex. 1887.)